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The concept of H -sets as introduced by Collatz in 1956 was very useful in univariate Chebyshev approximation by polynomials or Chebyshev spaces. In the multivariate setting, the situation is much worse, because there is no alternation, and H -sets exist, but are only rarely accessible by mathematical arguments. However, in Reproducing Kernel Hilbert spaces, H -sets are shown here to have a rather simple and complete characterization. As a byproduct, the strong connection of H -sets to Linear Programming is studied. But on the downside, it is explained why H -sets have a very limited range of applicability in the times of large-scale computing. [ABSTRACT FROM AUTHOR]

We study embeddings and norm estimates for tensor products of weighted reproducing kernel Hilbert spaces. These results lead to a transfer principle that is directly applicable to tractability studies of multivariate problems as integration and approximation, and to their infinite-dimensional counterparts. In an application we consider weighted tensor product Sobolev spaces of mixed smoothness of any integer order, equipped with the classical, the anchored, or the ANOVA norm. Here we derive new results for multivariate and infinite-dimensional integration. [ABSTRACT FROM AUTHOR]

We compare the Kolmogorov and entropy numbers of compact operators mapping from a Hilbert space into a Banach space. These general findings are then applied to embeddings between reproducing kernel Hilbert spaces and L ∞ ( μ ) . Here a sufficient condition for a gap of the order n 1 / 2 between the associated interpolation and Kolmogorov n -widths is derived. Finally, we show that in the multi-dimensional Sobolev case, this gap actually occurs between the Kolmogorov and approximation widths. [ABSTRACT FROM AUTHOR]

Abstract: The univariate Taylor formula without remainder allows to reproduce a function completely from certain derivative values. Thus one can look for Hilbert spaces in which the Taylor formula acts as a reproduction formula. It turns out that there are many Hilbert spaces which allow this, and they should be called Taylor spaces. They have certain reproducing kernels which are either polynomials or power series with nonnegative coefficients. Consequently, Taylor spaces can be spanned by translates of various classical special functions such as exponentials, rationals, hyperbolic cosines, logarithms, and Bessel functions. Since the theory of kernel-based interpolation and approximation is well-established, this leads to a variety of results. In particular, interpolation by shifted exponentials, rationals, hyperbolic cosines, logarithms, and Bessel functions provides exponentially convergent approximations to analytic functions, generalizing the classical Bernstein theorem for polynomial approximation to analytic functions. Finally, we prove sampling inequalities in Taylor spaces that allow to derive similar convergence rates for non-interpolatory approximations. [Copyright &y& Elsevier]

Abstract: The classical support vector machines regression (SVMR) is known as a regularized learning algorithm in reproducing kernel Hilbert spaces (RKHS) with a -insensitive loss function and an RKHS norm regularizer. In this paper, we study a new SVMR algorithm where the regularization term is proportional to -norm of the coefficients in the kernel ensembles. We provide an error analysis of this algorithm, an explicit learning rate is then derived under some assumptions. [Copyright &y& Elsevier]

Abstract: We propose an early stopping algorithm for learning gradients. The motivation is to choose “useful” or “relevant” variables by a ranking method according to norms of partial derivatives in some function spaces. In the algorithm, we used an early stopping technique, instead of the classical Tikhonov regularization, to avoid over-fitting. After stating dimension-dependent learning rates valid for any dimension of the input space, we present a novel error bound when the dimension is large. Our novelty is the independence of power index of the learning rates on the dimension of the input space. [Copyright &y& Elsevier]

Abstract: The main purpose of the present paper is to employ spherical basis functions (SBFs) to study uniform distribution of points on spheres. We extend Weyl''s criterion for uniform distribution of points on spheres to include a characterization in terms of an SBF. We show that every set of minimal energy points associated with an SBF is uniformly distributed on the spheres. We give an error estimate for numerical integration based on the minimal energy points. We also estimate the separation of the minimal energy points. [Copyright &y& Elsevier]

Abstract: The tractability of multivariate problems has usually been studied only for the approximation of linear operators. In this paper we study the tractability of quasilinear multivariate problems. That is, we wish to approximate nonlinear operators  that depend linearly on the first argument and satisfy a Lipschitz condition with respect to both arguments. Here, both arguments are functions of  variables. Many computational problems of practical importance have this form. Examples include the solution of specific Dirichlet, Neumann, and Schrödinger problems. We show, under appropriate assumptions, that quasilinear problems, whose domain spaces are equipped with product or finite-order weights, are tractable or strongly tractable in the worst case setting. This paper is the first part in a series of papers. Here, we present tractability results for quasilinear problems under general assumptions on quasilinear operators and weights. In future papers, we shall verify these assumptions for quasilinear problems such as the solution of specific Dirichlet, Neumann, and Schrödinger problems. [Copyright &y& Elsevier]